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Dynamic Analysis of the System
of the Moored-spherical-buoy
Oceanology International
23-25 OCT 2018, Qingdao, China
Prof. Xiangqian Zhu
Department of Mechanical Engineering,
Shandong University
Shandong University2/24
Modeling of Cable
1. Background
2. Formulation
3. Element reference frame
4. Advantages of new ERF
Modeling of Spherical Buoy with Cable
1. Geometrical modeling
2. Formulations of loads
3. Spherical buoy with cables
Summary
Contents
Shandong University3/24
Dy
na
mic
An
aly
sis No. ① ②
Tension O O
Strain X O
Damping X O
Hydrodyn X O
Efficiency ●● ●
Accuracy ● ●●
Fastening Floating PlatformQ
ua
si-s
tati
c A
naly
sis
① ②
③
Computer Technology
Composite Materials
Small Structural
Accuracy Requirement
Dynamics of cables should be
considered in analyzing floating
structures with mooring cables !
Finite difference model
tn
bLumped mass model
A B
T T & S
ODE PDE
Low- order High-order
Robust Sensitive
A
B
Towing Marine Vehicles
Development New Technology
Background
Shandong University4/24
cos sin sin sin cos
0 cos sin
sin cos sin cos cos
i i i i i
i i
i i i i i
1,1 ,1 1,3 ,3
1,3 ,3
1,2 ,2
1,1 ,1
1,2 ,2
atant 2( , )
atant 2 ( ), , cos sincos
atant 2 ( ), , cos sinsin
i i i i i
g g g g
i i
g gi i i i i
g g i
i i
g gi i i i i
g g i
N N N N
N NN N if
N NN N if
Euler Angles
Gimbal lock
Calculation of
rotation angles
Frenet Frame
Non-zero curvature
Looseness and Tightness
X-traversing with
X-current
Y-traversing with
Y-current
X-displacement of the 1st node
1. Deal with singular problems
2. Express loads efficiently
A new Element frame
To consider the influence of thedynamics of cables on thesystem of floating structureswith mooring cables, the cablemodeling is established basedon a new element framethrough which not only thecable modeling is competent tomodel the situations where thesingular problems are generatedby the Euler angles and Frenetframe, but also thehydrodynamic loads acting oncables are expressed efficiently.
Lumped mass model
“Shape” ---static
“Loads” ---dynamics
“Environment”Modes of action of loads
Background
Shandong University5/24
1
1
2
i i i
g g g
i i
g gi
g
R f i
g g g
E N N
N NV
V V V
& &
cos( , ) cos( , ) cos( , )
cos( , ) cos( , ) cos( , )
cos( , ) cos( , ) cos( , )
x x x
iy y y
z z z
u u u u u u
A u u u u u u
u u u u u u
|| ||
|| ||
i
gi
i
g
i R
gi
i R
g
i i i
Ez
E
z Vx
z V
y z x
%
%
%
[ , , ]i i i iA x y z
Formulation of cable
Shandong University6/24
1 1 1 1 1
1 1( )
2 2
1 1( )
2 2
i i i i i i i
I g b b D W
i i i i i i
b b D W ext
M N A T D F F
A T D F F F
&&
Governing Equation for Modeling of Cable
Loads acting on
ith Element
Loads acting on
(i-1)th Element
Loads
acting on
ith Node
Loads acting on
ith Node.
Formulation of cable
Shandong University7/24
Parameters Magnitude Unit
X-directional wave amplitude 1.2 m
X-directional wave period 8 S
Velocity of current [0; 1; 0] m/s
Density of fluid 1025 kg/m3
Stiffness coefficient 8.70e7 N/m
Damping coefficient 3.30e6 Ns/m
Parameters Magnitude Unit
Diameter 0.03 m
Density 7800 kg/m3
Elastic modulus 2.0e11 N/m
Damping coefficient 1.0e4 Ns/m
Normal drag coefficient 1
Tangential drag coefficient 0.01
Added mass coefficient 1
Position of 1st node [10; 0; -10] m
Position of 11th node [0; 0; -20] m
cos sin sin sin cos
0 cos sin
sin cos sin cos cos
i i i i i
i i i
i i i i i
A
cos( , ) cos( , ) cos( , )
cos( , ) cos( , ) cos( , )
cos( , ) cos( , ) cos( , )
x x x
iy y y
z z z
u u u u u u
A u u u u u u
u u u u u u
Euler angles
(XYZ) set
Set Z is zero
Frenet frame
&
New Element
frame
750 NFrenet o
Euler ─◁─
New ERF •
Cable modeling---ERF
Shandong University8/24
Therefore, the following characteristics of the Euler angles can be concluded from above results:
(1) rotation angles of one element are only related to the direction of the z-axis that indicates the orientation of the said element;
(2) the RTM of one element is merely identified by the orientation of the said element.
1st element 2nd element 9th element 10th element
Euler angles
Cable modeling---ERF
Shandong University9/24
Therefore, characteristics of the Frenet frame are:
(1) the components of the RTM change to ensure that the z-axis is tangential to the cable and that the x-axis is normal to the cable;
(2) the ERF for one element is defined by the said element and two adjacent elements together; and
(3) the normal vector is easily undefined when the second derivative of the spline function of the cable becomes very small.
1st element 2nd element 9th element 10th element
Frenet Frame
Cable modeling---ERF
Shandong University10/24
1st element 2nd element 9th element 10th element
The new ERF is generated on a basis of the following assumptions:
(1) the vector of the relative velocity is non-zeros; and
(2) the vector of the relative velocity is non-collinear with the vector of the element orientation.
New
Element Frame
Cable modeling---ERF
Shandong University11/24
,1 ,3
,3
,2
,1
,2
tan 2 ,
tan 2 , , cos sincos
tan 2 , , cos sinsin
i i ig g
igi i i i
g i
igi i i i
g i
a E E
Ea E if
Ea E if
Move X-directionally with X-directional current
Move Y-directionally with Y-directional current
X-directional difference is 0, so the rotation angle θ is
180°, which results in the denominator of the atant2
function being 0. This forces the rotation angle ϕ being
pi/2 or –pi/2 easily in spite of the values in the front
component of the atan2 function. Simulation results
indicate that the rotation angle ϕ is pi/2 or –pi/2
interactively
Euler *
New ERF o
ϕ angle
Cable modeling---singularity
Shandong University12/24
Components of the RTM by Frenet Frame
in singularity cases(the 1st element)
X-disp. of the 1st and 2nd nodes
X-disp. of the 1st node Z-disp. of the 1st node
Components of the RTM by Frenet Frame
in singularity cases(the 10th element)
Euler ─◁─
New ERF •
Step size 1e-5
Frenet Frame is invalid
Cable modeling---singularity
Shandong University13/24
3 drag forces in element
frame with relative
velocity in ERF
3 drag forces
in IRF
2 drag forces in element
frame with relative
velocity in IRF
Frenet frame New element frame
Relative velocity in
ERF from the IRF
3 drag forces
in IRF
, ,
, ,
, ,
1
2
1
2
2
i i i R x R xx n f b b
i i i R y R yy n f b b
i i i R z R zz f f b b
F C d l V V
F C d l V V
F C d l V V
,
2 2, ,
,
2 2, ,
2,
1
2
1
2
1sgn( )
2
R xi i i R bx n f p b
R x R yb b
R yi i i R by n f p b
R x R yb b
i R z i i Rz b n f q b
VF C d l f V
V V
VF C d l f V
V V
F V C d l f V
2 3 4 5
'
0.5 0.1cos 0.1sin 0.4cos 2 0.11sin 2
0.01 2.008 0.3858 1.9159 4.1615 3.5064 1.1873
cos 02
p
q
Rb
Rb
f
f
V za
V z
21
2
2
%i i i iy n f g
i i i T i T iz f f g g
F C d l zV
F C d l z V z VNG
NB
NBL
,
,
sin
cos
i y ib g
i z ib g
V V
V V
,
,
%i y ib g
i z T ib g
V zV
V z V
where
Hydrodynamic
drag force Relationship
Relative Velocity &
Unit-axes of frame
Empirical
formulation with
Loading function
parameters
Cable modeling---drag force
Shandong University14/24
X-directional displacement Y-directional displacement
X-directional Hydrodynamic loads Y-directional Hydrodynamic loads
Eta angle
Loading function fp
The fp is negative when the Eta is close to the 0.1, which
changes the direction of the hydrodynamic drag force.
Therefore, the hydrodynamic drag force based on the
loading functions is incorrect when the cable is pulled
taut. The formulation based on the new element frame is
appropriate in the case of vertically taut cables.
Step size 0.001
Zhu Xiang Qian, Yoo Wan Suk*. Suggested New Element Reference Frame for Dynamic Analysis of Marine
Cables;Nonlinear Dynamics;87(1),p489–501; 2017.
Cable modeling---drag force
Shandong University15/24
Meteorology monitoringNavigation
Nonlinear elevation
of the surface wave
Nonlinear
spherical surface
Closed curve
Polyhedral mesh
1. Closed curve is co-plane.
2. Morison equation is valid.
3. Rotational motions of the buoy are ignored.
4. All the loads acting on the 1st node of the cable.
Efficient modeling is preferred !
Assumptions
The size of the spherical buoy is
small, so the dynamics of the cable
have obvious influence on the
motions of the system.
Anti-shark cable net barrier
Spherical buoy
Shandong University16/24
The cross-section is determined bythree points, a, b and c, whichlocate on the edge of the concentriccircle.
The cross-section is determined by three
points, a, b and c, which locate on the
edge of an assumed concentric circle.
Concentric
circle
22 2 2
0 0 0
1,3
23 2 2 2
0 0 0
1,3
sin ( )3
4sin ( )
3 3
R
g
sR
g
d d r dr SH R SH
if PH N RV
R d d r dr SH R SH
if PH N R
Submerged Volume
Cone
1,1 ,1 ,1 ,1
1,2 ,2 ,2 ,2
1,3 ,3 ,3 ,3 | |
buoy buoy buoyext B A D
buoy buoy buoyext B A D
buoy buoy buoy buoyext B A D
F F F F
F F F F
F F F F m g
Hydrodynamic Loads
Hydrodynamic loads are
expressed on the 1st node of the
cable based on the submerged
volume
Error
x, y
z cos( ) cos( ) x x x y y ya g a gk X t k Y t
Spherical buoy
Shandong University17/24
Spherical Buoy Parameters Magnitude Unit
Radius 1 m
Mass 500 kg
Radius of the concentric circle 0.8 m
Drag coefficient 1
Added mass coefficient 1
Ocean State Parameters Magnitude Unit
X-directional wave amplitude 0.7 m
Y-directional wave amplitude 6.4 m
X-directional wave period 1.2 s
Y-directional wave period 8 s
Velocity of current [1; 0; 0] m/s
Density of fluid 1025 kg/m3
Stiffness of seabed 87 MN/m
Damping of seabed 33 MN • s/m
Efficiency Comparison
Parameters ProteusDS NM Code
Simulation time 20 s 20 s
Integrator RK 4 RK 4
Step size 1e-4 1e-4
Real time 2655 s 1699 s
Rate 1 64 %
Cable Property Parameters Magnitude Unit
Diameter 0.03 m
Density 1570 kg/m3
Elastic modulus 2.38e9 N/m
Damping coefficient 1.0e4 Ns/m
Normal drag coefficient 1
Tangential drag coefficient 0.01
Added mass coefficient 0.5
Position of 1st node [10; 0; -10] m
Position of 11th node [0; 0; -30] m
Spherical buoy
Shandong University18/24
Surge motion of buoy Sway motion of buoy Heave motion of buoy
Frequency of Surge Frequency of Sway Frequency of Heave
Tension within cable Frequency of Tension Enlarged heave motionZhu Xiang Qian, Yoo Wan Suk*. Numerical modeling of a spherical buoy moored by a cable in threedimensions;Chinese Journal of Mechanical Engineering;29(3);p588–597; 2016.
Spherical buoy
Shandong University19/24
Ocean State Parameters Magnitude Unit
X-directional wave amplitude 0.7 m
Y-directional wave amplitude 1.2 m
X-directional wave period 6.4 s
Y-directional wave period 8 s
Cable Property Parameters Magnitude Unit
Diameter 0.03 m
Density 3570 kg/m3
Elastic modulus 2.38e9 N/m
Damping coefficient 1.0e4 Ns/m
Normal drag coefficient 1
Tangential drag coefficient 0.01
Added mass coefficient 0.5
Position of the top node [0; 0; 0] m
Bottom node of cable A [-30; 0; -20] m
Bottom node of cable B [30; 0; -20] m
Spherical buoy
Shandong University20/24
21 3
Tension 1st element Cable A Tension 1st element Cable B Wave Amplitude
Tension 5th element Cable A Tension 5th element Cable B
Tension Freq. of 1st element A Tension Freq. 1st element B Surge Freq. of buoy
Surge of buoy
Spherical buoy
Shandong University21/24
Sway Freq. of buoy
Sway of buoy Heave of buoy
Heave Freq. of buoy
The Roll and Pitch motions of
the buoy is ignored in the
numerical modeling. It assume
the ligature between the center of
the buoy and the node is
vertical upward during the
simulation. While, the buoy has
the rotational motions in
ProteusDS, so the buoy can
avoid the wave crest by a short
time. This phenomenon enable a
time delay to exist in the motion
of the buoy.
All the loads acting on the buoy,
including the Hydrostatic and
Hydrodynamic loads, expressed
on the bottom of the buoy, which
is the 1st node of the cable in the
NM Code. The hydrodynamic
loads obtained by the proposed
modeling are smaller than those
obtained by the Polygonal mesh.
Spherical buoy
Shandong University22/24
Freq. of x-displacement of the
5th nodes in Cable B
Freq. of y-displacement of the 5th
nodes in Cable B
Freq. of z-displacement of the
5th nodes in Cable B
X-displacement of the 5th nodes
in Cable A and B
Y-displacement of the 5th nodes
in Cable A and B
Z-displacement of the 5th nodes
in Cable A and B
Zhu Xiang Qian,Yoo Wan Suk*. Dynamic Analysis of a Floating Spherical Buoy Fastened by Mooring Cables;Ocean Engineering;121,p462–471. 2016;
Spherical buoy
Shandong University23/24
Summary
• By means of the relative velocity, a new element reference frame is constructed for
modeling marine cable.
• The hydrodynamic loads are expressed efficiently, but also many singular problems can be
overcome easily.
• By means of the special coordinate, a analytical method is proposed to characterize the
hydrodynamics acting on the spherical buoy.
• The relationships between the motions of the system and propagating waves are studied,
and the simulation results are verified by commercial software ProteusDS.
Summary
Acknowledgement: The DSA Ltd. for providing the software ProteusDS
Shandong University24/24
Other Publications
Zhu Xiang Qian,Yoo Wan Suk*. Verification of a Numerical Simulation Code for Underwater Chain Mooring;Archive ofMechanical Engineering;63(2);p231-244; 2016
Zhu Xiang Qian,Yoo Wan Suk*.Numerical Modeling of a Spar Platform Tethered by a Mooring Cable;Chinese Journal ofMechanical Engineering;28(4),p785–792;2015.
Zhu Xiang Qian,Yoo Wan Suk*. Flexible dynamic analysis of an offshore wind turbine installed on a floating spar platform;Advances in Mechanical Engineering;8(6);p1-11; 2016
Jiang Zhiyu, Zhu Xiang Qian*, Hu weifei. Modeling and Analysis of Offshore Floating Wind Turbines, Chapter 9,《Advanced
Wind Turbine Technology》, Springer International Publishing, 2018
Thank you very much!
Paper list